What is the probability of all three companies going bankrupt?

Question

So I had a question on a test which I can't stop thinking about.

The question is: say that there are three companies $(A,B,C)$ that are independent of each other. The probability of company $B$ going bankrupt is $0.20$. The probability of both $A$ and $C$ going bankrupt is $0.07$, and the probability of $B$ and $C$ going bankrupt is $0.08$. If we know that at least two of the companies have gone bankrupt, what is the probability of all three of them having gone bankrupt?

$$P(B) = 0.2$$ $$P(A\cap C) = 0.07$$ $$P(B\cap C) = 0.08$$

Isn't this just $P(\textrm{all three bankrupt}\; |\; \textrm{at least two of the companies having gone bankrupt})$?

And if we use the rule $P(A\;|\;B) = P(A)$ for independent events (if we can?), shouldn't the answer to this question just be $P(\textrm{All three bankrupt})$ Which would just be $P(A) \cdot P(B) \cdot P(C)$?

Is there something I've done wrong? This was a question on an exam for a statistics basic course at UNI.

Just confused if this is really the correct answer or if its supposed to be way harder?

Answer 1

$P(\textrm{all three bankrupt}\; |\; \textrm{at least two bankrupt})$.......if we use the rule $P(A\;|\;B)= P(A)$ for independent events (if we can?), shouldn't the answer to this question just be $P(\textrm{all three bankrupt})?$

But the event that all three have gone bankrupt is not independent of the event that at least two have gone bankrupt.

$$P(\text{all three bankrupt}|\text{at least two bankrupt})\\= \frac{P(\text{all three bankrupt})}{P(\text{at least two bankrupt})}\\= \frac{P(A\cap B\cap C)}{P(A\cap B)+P(B\cap C)+P(C\cap A)-2P(A\cap B\cap C)}\\= \frac{0.014}{0.035+0.08+0.07-2(0.014)}\\=8.92\%$$

Answer 2

Let $X$ denote the number of companies that go bankrupt.

Then to be found is:$$P(X=3|X\geq2)=\frac{P(X=3,X\geq2)}{P(X\geq2)}=\frac{P(X=3)}{P(X\geq 2)}$$ Here: $$P(X=3)=P(A\cap B\cap C)$$ and:$$P(X\geq2)=P(X=3)+P(X=2)=$$$$P(A\cap B\cap C)+P(A\cap B\cap C^c)+P(A\cap B^c\cap C)+P(A^c\cap B\cap C$$I leave the rest to you.